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%I #21 Apr 03 2023 14:31:52
%S 0,3,22,126,652,3190,15060,69356,313624,1398438,6166660,26948548,
%T 116888232,503811516,2159864392,9216445080,39168381488,165864540934,
%U 700151508324,2947120122068,12373581565960,51831196048212,216659135089496,903925011410536
%N a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(n,j)*binomial(n,k)*max(j,k).
%H G. C. Greubel, <a href="/A100511/b100511.txt">Table of n, a(n) for n = 0..1000</a>
%H M. Klamkin, ed., <a href="https://doi.org/10.1137/1.9781611971729">Problems in Applied Mathematics: Selections from SIAM Review</a>, SIAM, 1990; see pp. 127-129.
%F a(n) = n*2^(2*n-1) + (n/2)*binomial(2*n, n). [Typo corrected by _Ognjen Dragoljevic_, Dec 26 2017]
%F From _G. C. Greubel_, Apr 01 2023: (Start)
%F G.f.: x*(2 + sqrt(1-4*x))/(1-4*x)^2.
%F E.g.f.: x*(2*exp(4*x)+ exp(2*x)*(BesselI(0, 2*x) + BesselI(1, 2*x))). (End)
%t Table[n*(4^n +(n+1)*CatalanNumber[n])/2, {n,0,40}] (* _G. C. Greubel_, Apr 01 2023 *)
%o (PARI) a(n) = n*2^(2*n-1) + (n/2)*binomial(2*n, n); \\ _Michel Marcus_, Dec 26 2017
%o (Magma) [n*(4^n +(n+1)*Catalan(n))/2: n in [0..40]]; // _G. C. Greubel_, Apr 01 2023
%o (SageMath) [n*(4^n +binomial(2*n,n))/2 for n in range(41)] # _G. C. Greubel_, Apr 01 2023
%Y Cf. A000108, A002457, A033504.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Nov 24 2004
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